1. **State the problem:** Simplify the expression $2x^2 \cdot 2x$. 2. **Recall the rule for multiplying coefficients and variables:** When multiplying terms with the same base, add the exponents: $x^a \cdot x^b = x^{a+b}$. Also multiply the numerical coefficients. 3. **Multiply the coefficients:** $2 \cdot 2 = 4$. 4. **Multiply the variables:** $x^2 \cdot x = x^{2+1} = x^3$. 5. **Combine the results:** The expression simplifies to $$4x^3$$. 6. **Final answer:** $$4x^3$$.